Classification of certain types of maximal matrix subalgebras

Abstract Let M n ( K ) denote the algebra of n × n matrices over a field K of characteristic zero. A nonunital subalgebra N ⊂ M n ( K ) will be called a nonunital intersection if N is the intersection of two unital subalgebras of M n ( K ) . Appealing to recent work of Agore, we show that for n ≥ 3 , the dimension (over K ) of a nonunital intersection is at most ( n − 1 ) ( n − 2 ) , and we completely classify the nonunital intersections of maximum dimension ( n − 1 ) ( n − 2 ) . We also classify the unital subalgebras of maximum dimension properly contained in a parabolic subalgebra of maximum dimension in M n ( K ) .